A156128 - OEIS

%I #50 Sep 08 2022 08:45:41

%S 1,6,72,1080,18144,326592,6158592,120092544,2401850880,48997757952,

%T 1015589892096,21327387734016,452796847276032,9702789584486400,

%U 209580255024906240,4558370546791710720,99747873141559787520,2194453209114315325440,48508965675158549299200

%N a(n) = 6^n * Catalan(n).

%C Number of Dyck n-paths with two types of up step and three types of down step. - _David Scambler_, Jun 21 2013

%H Vincenzo Librandi, <a href="/A156128/b156128.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = 6^n * A000108(n).

%F From _Gary W. Adamson_, Jul 18 2011: (Start)

%F a(n) is the upper left term in M^n, M = an infinite square production matrix:

%F   6, 6, 0, 0, 0, 0, ...

%F   6, 6, 6, 0, 0, 0, ...

%F   6, 6, 6, 6, 0, 0, ...

%F   6, 6, 6, 6, 6, 0, ...

%F   ... (End)

%F E.g.f.: KummerM(1/2, 2, 24*x). - _Peter Luschny_, Aug 26 2012

%F G.f.: c(6*x) with c(x) the o.g.f. of A000108 (Catalan). - _Philippe Deléham_, Nov 15 2013

%F a(n) = Sum{k=0..n} A085880(n,k) * 5^k. - _Philippe Deléham_, Nov 15 2013

%F G.f.: 1/(1 - 6*x/(1 - 6*x/(1 - 6*x/(1 - ...)))), a continued fraction. - _Ilya Gutkovskiy_, Aug 08 2017

%F Sum_{n>=0} 1/a(n) = 588/529 + 864*arctan(1/sqrt(23)) / (529*sqrt(23)). - _Vaclav Kotesovec_, Nov 23 2021

%F Sum_{n>=0} (-1)^n/a(n) = 564/625 - 432*log(3/2) / 3125. - _Amiram Eldar_, Jan 25 2022

%F D-finite with recurrence (n+1)*a(n) +12*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Mar 21 2022

%p A156128_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;

%p for w from 1 to n do a[w] := 6*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od;convert(a,list)end: A156128_list(16); # _Peter Luschny_, May 19 2011

%t Table[CatalanNumber[n]6^n, {n, 0, 16}] (* _Alonso del Arte_, Jul 19 2011 *)

%o (Magma) [6^n*Catalan(n): n in [0..20]]; // _Vincenzo Librandi_, Jul 19 2011

%Y Cf. A000108, A005159, A085880, A151374, A151403, A156058.

%Y Column k=6 of A290605.

%K easy,nonn

%O 0,2

%A _Philippe Deléham_, Feb 04 2009